Gemara

(Rabanan of the Beis Medrash): One may fast for part of a day. If he finished, he can say Tefilas Ta'anis (Aneinu, which we say on fast days).

Ta'anis 11b (Rav Huna): If one spent the night in his fast, he does not say Tefilas Ta'anis.

Question (Rav Yosef): Does Rav Huna hold that one may not fast hours (less than a full day)? Or does he hold that one may fast hours, but one who does so does not say Tefilas Ta'anis?

Objection (Abaye): Perhaps he holds that one may fast hours, and one who does so says Tefilas Ta'anis, but here he cannot because he did not accept to fast the night hours beforehand!

12a (Rav Chisda): One may fast for part of a day, i.e. if he does not eat at all until evening.

Objection (Abaye): That is a full fast! Rather, Rabanan teaches about one who reconsidered.

(Rav Chisda): Any fast on which the sun did not set, is not a fast.

If he intends to eat before this, he may not say Aneinu.

Eruvin 41a (Beraisa - R. Yehudah): Once, Tishah b'Av fell on Erev Shabbos. R. Akiva ate an egg, to show the Halachah to Talmidim.

R. Yosi says, he must complete the fast. (R. Akiva was sick.)

Chachamim: One may not enter Shabbos amidst affliction!

(Mar Zutra): The Halachah is, one completes the fast.

Pesachim 54b (Shmuel): Bein ha'Shemashos of Tish'ah b'Av (and every Ta'anis Tzibur) is permitted.

(Rava and R. Yochanan): It is forbidden.

Rishonim

Rambam (Hilchos Ta'anis 1:13): One may fast hours, if he does not eat for the rest of the day. If one was busy and did not eat until midday or nine hours of the day, and he decided to fast the rest of the day, he fasts those hours and says Aneinu, for he accepted it before the hours of the fast. Similarly, if he ate and drank and began to fast the rest of the day, this is a fast of hours.

Rebuttal (Ra'avad): The Gemara said 'we must say that he did not eat the entire day', and 'any fast on which the sun did not set, is not a fast.' The former excludes one who ate and decided to fast the rest of the day. The latter invalidates an acceptance to fast until midday.

Rosh (Ta'anis 1:12): The Yerushalmi connotes like the Rambam.It learns from R. Yochanan and R. Yonah that one may fast hours, and from Rav, who allows to borrow and repay a fast, i.e. this day that he began to fast is called a fast, and he (need not - the Mordechai and others say that the text should say 'must') repay, for he accepted a full day. Avi ha'Ezri rejected the Rambam, and said that the Bavli is unlike the Yerushalmi. I say that we can resolve them. The Yerushalmi does not discuss Tefilas Ta'anis. Rather, it teaches that one must fulfill his vow, but he does not say Tefilas Ta'anis unless he fasts a full day. Rav said that one may borrow and repay a fast, i.e. even if he accepted it, he can borrow. Since he fasted part of the day, this is called a (partial) Ta'anis, and we do not say that eating was a negation of his vow. If one may not fast hours, if he accepted to fast until midday, and completed the fast, he does not say Aneinu, and if he was obligated to fast, he was not Yotzei. If so, it is an invalid acceptance, and he may eat.

Korban Nesanel (20): This is if he said 'I am in a fast.' If he said 'it is upon me not to eat tomorrow', this is a vow, and he may not eat, like it says in Shulchan Aruch.

Rosh (ibid): Rashi explained that a fast of hours is if he accepted to fast from midday until evening, and he did not eat in the morning. The Ramban says that it is even if he did not accept from Minchah on the previous day, since he accepted it before the hours of the fast. This is astounding. Why is this different than a full day fast, which must be accepted at Minchah the day before, and if he did not, it is not a fast?! We can say that there, he needs a full day fast, so he must accept it before the fast begins. If so, if one did not intend to fast, and at night he decided to fast, it is a fast. He is no worse than one who decided to fast after six hours. However, Abaye says oppositely! Rather, every fast must be accepted at Minchah the day before. Rav Chisda taught that any fast on which the sun did not set, is not a fast. Primarily he discusses one who accepted a fast Stam. He must fast until Tzeis ha'Kochavim. Even though we say that from sunset until Tzeis ha'Kochavim is the time to walk five Mil (90 minutes), R. Tam says that here Rav Chisda discusses Sof Shki'ah, i.e. Tzeis ha'Kochavim. We conclude that one must accept a fast from the day before, and one must finish it until the evening.

Poskim

Shulchan Aruch (OC 562:1): Any fast on which the sun did not set, i.e. he did not complete it until Tzeis ha'Kochavim (Rema - when three medium size stars are visible, or when the moon shines strongly and illuminates on the ground) is not a fast. If he intends to eat before this, he may not pray Aneinu.

Beis Yosef (DH Omar): Hagahos Maimoniyos and Semak say that many eat at sunset. However, Tosfos says in the name of R. Elchanan that the custom is to fast until Tzeis ha'Kochavim even for an individual's fast. 'Shki'ah' refers to the end of Shki'ah, i.e. Tzeis ha'Kochavim. Even though the Gemara says that Bein ha'Shemashos begins after Shki'ah, which is before Tzeis ha'Kochavim, since most people do not know when is the end of Shki'ah, the custom is to complete the fast until Tzeis ha'Kochavim. Pesachim 54a connotes that we are stringent about the Safek of Bein ha'Shemashos for Tish'ah b'Av, but not for others fasts. We can say that Bein ha'Shemashos at the start of the day is more lenient, but Bein ha'Shemashos at the end of the day is stringent, and one must wait until it is Vadai night, i.e. Tzeis ha'Kochavim. Hagahos Maimoniyos learns from the Yerushalmi that when the moon shines and illuminates on the ground, we know that the sun set. This is relevant to fasts and Shabbos.

Beis Yosef (DH u'Mah she'Chosav): The Gemara, Rambam, Rosh and Ran connote that 'it is not a fast' means that one does not say Tefilas Ta'anis. We say that one does not say Aneinu if the sun did not set on the fast. It seems that this includes if he did not yet eat, but he intends to eat before Shki'ah.

Gra (1): Chachamim (Eruvin 41a) said that one may not complete the fast on Erev Shabbos, lest he enter Shabbos amidst affliction. This shows that the fast ends at Sof Shki'ah. (If not, one could complete the fast and eat before Shabbos began!) Amora'im argue about whether Bein ha'Shemashos of Tish'ah b'Av is permitted. This implies that Bein ha'Shemashos of other fasts is surely permitted, i.e. even at the end of the day.

Kaf ha'Chayim (2): We are stringent about all the fasts because not everyone knows what is the end of the (second) Shki'ah.

Mishnah Berurah (1): Here we discuss the four fasts (Tzom Gedalyah, 10th of Teves, Ta'anis Esther and 17th of Tamuz) or a fast that an individual accepted on himself.

Mishnah Berurah (3): After Shabbos or Yom Kipur, we are stringent to wait for three small stars near each other. Here, three medium stars suffice, even if they are scattered.

Kaf ha'Chayim (4): Also, one must add to Shabbos and Yom Kipur, but one need not add to fasts.

Rema: However, our custom is to pray Aneinu even though we do not finish until Tzeis ha'Kochavim. Some Rabanan hold like this. I say that only an individual says Aneinu in Shome'a Tefilah, for in any case one may add in this Berachah, but a Shali'ach Tzibur (in the repetition) says Aneinu only if they finish. This is the custom.

Taz (1): The Rashba said that the Rambam retracted. How can one say 'on this day of our fast' if it is not a fast?! The Bach said that the Rambam, Ramban, Ba'al ha'Ma'or and Ran hold that it suffices to accept before a fast of hours (he need not accept the day before). This is wrong. Rather, if one fasts an entire day, it suffices to accept in the middle! One may rely on the custom, but one who omits Aneinu when he does not fast the entire day avoids a Safek Berachah, i.e. a Hefsek in Shome'a Tefilah. One should omit the words 'day of our fast.'

Mishnah Berurah (7): Bigdei Yesha and Machatzis ha'Shekel say that he need not omit 'day of our fast.'

Kaf ha'Chayim (8): Mishbetzos Zahav (1) says that even if he did not accept to fast, and even for a fast of hours, he says Aneinu without 'day of our fast.' It is better to do so in Elokai Netzor.

Magen Avraham (2): If one must stipulated that he will fast until Minchah Gedolah (a half hour after midday), he can say Aneinu. If he accepted a fast Stam, he must complete it.

Mishnah Berurah (6): Even the Rema does not allows saying Aneinu if he already ate.